Theorem bnj1071 35674

Description: Technical lemma for bnj69 35707. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Hypothesis

Ref	Expression
bnj1071.7	⊢ 𝐷 = (ω ∖ {∅})

Assertion

Ref	Expression
bnj1071	⊢ (𝑛 ∈ 𝐷 → E Fr 𝑛)

Proof of Theorem bnj1071

Step	Hyp	Ref	Expression
1		bnj1071.7	. . 3 ⊢ 𝐷 = (ω ∖ {∅})
2	1	bnj923 35466	. 2 ⊢ (𝑛 ∈ 𝐷 → 𝑛 ∈ ω)
3		nnord 7906	. 2 ⊢ (𝑛 ∈ ω → Ord 𝑛)
4		ordfr 6392	. 2 ⊢ (Ord 𝑛 → E Fr 𝑛)
5	2, 3, 4	3syl 19	1 ⊢ (𝑛 ∈ 𝐷 → E Fr 𝑛)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2147   ∖ cdif 4048  ∅c0 4281  {csn 4588   E cep 5554   Fr wfr 5605  Ord word 6376  ωcom 7898

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-uni 4871  df-tr 5216  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-ord 6380  df-on 6381  df-om 7899

This theorem is used by:  bnj1030  35684  bnj1133  35686